Question 74

# What will be the measure of the acute angle formed between the hour hand and the minute hand at 6:43 a.m.?

Solution

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

$$\frac{360^\circ total}{60 minutes total}=6^\circÂ per minute$$

The minute hand on the clock will point at 43 minutes, allowing us to calculate it's position on the circle.

(43 min)(6)=$$258^\circ$$

Since there are 12 hours on the clock, each hour mark is 30 degrees.

$$\frac{360^\circ total}{12 hours total}=30^\circ per hour$$

We can calculate where the hour hand will be at 6:00.

$$(6 hr)(30)=180^\circ$$

However, the hour hand will actually be between the 6 and 7, since we are looking at 6:43 rather than an absolute hour mark. 43 minutes is equal to $$\frac{43}{60}$$th of an hour. Use the same equation to find the additional position of the hour hand.

$$180^\circ + \frac{43}{60} \times 30 = 201.5^\circ$$

We are looking for the acute angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

Required answer = $$258^\circ - 201.5^\circ = 56.5^\circ$$

So, the answer would be option b)$$56.5^\circ$$.

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